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Find the length and width of a rectangle that has the given area and a minimum perimeter. Area: 5A square centimeters cm (smaller value) cm (larger value)

Find the length and width of a rectangle that has the given area and a minimum perimeter. Area: 5A square centimeters cm (smaller value) cm (larger value)

Calculus: Early Transcendentals
Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
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### Rectangle Area and Perimeter Problem **Problem Statement:** Find the length and width of a rectangle that has the given area and a minimum perimeter. **Given:** - **Area:** \( 5A \) square centimeters **To determine:** - Dimensions of the rectangle (length and width) that satisfy the given area and yield a minimum perimeter. **Steps to Follow:** 1. **Identify the relationship between length and width in terms of the area:** - Given: \( \text{Area} = 5A \) square centimeters - Let \( l \) be the length and \( w \) be the width of the rectangle. - Therefore, \( l \times w = 5A \) 2. **Determine the dimensions that minimize the perimeter:** - The perimeter \( P \) of a rectangle is given by \( P = 2(l + w) \). - To minimize the perimeter for a given area, the rectangle should be as close to a square as possible. - Hence, for this problem, the length and width will be: \( l = w = \sqrt{5A} \). **Required Input Boxes:** - Input box for the smaller value (width or length): \( \sqrt{5A} \) cm (smaller value) - Input box for the larger value (length or width): \( \sqrt{5A} \) cm (larger value) This problem helps in understanding how the dimensions of geometric shapes impact their perimeter given a fixed area, specifically in optimizing the shape to achieve certain conditions.
Transcribed Image Text:### Rectangle Area and Perimeter Problem **Problem Statement:** Find the length and width of a rectangle that has the given area and a minimum perimeter. **Given:** - **Area:** \( 5A \) square centimeters **To determine:** - Dimensions of the rectangle (length and width) that satisfy the given area and yield a minimum perimeter. **Steps to Follow:** 1. **Identify the relationship between length and width in terms of the area:** - Given: \( \text{Area} = 5A \) square centimeters - Let \( l \) be the length and \( w \) be the width of the rectangle. - Therefore, \( l \times w = 5A \) 2. **Determine the dimensions that minimize the perimeter:** - The perimeter \( P \) of a rectangle is given by \( P = 2(l + w) \). - To minimize the perimeter for a given area, the rectangle should be as close to a square as possible. - Hence, for this problem, the length and width will be: \( l = w = \sqrt{5A} \). **Required Input Boxes:** - Input box for the smaller value (width or length): \( \sqrt{5A} \) cm (smaller value) - Input box for the larger value (length or width): \( \sqrt{5A} \) cm (larger value) This problem helps in understanding how the dimensions of geometric shapes impact their perimeter given a fixed area, specifically in optimizing the shape to achieve certain conditions.
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